Question

if ( 6x leq g(x) leq 3x^{4}-3x^{2}+6 ) for all ( x ), evaluate ( lim _{x
ightarrow 1} g(x) ).

Explanation:

Step1: Find $\lim_{x

ightarrow1}6x$
Substitute \(x = 1\) into \(6x\).
\(\lim_{x
ightarrow1}6x=6\times1 = 6\)

Step2: Find $\lim_{x

ightarrow1}(3x^{4}-3x^{2}+6)$
Use the sum - difference rule \(\lim_{x
ightarrow a}(u(x)\pm v(x)\pm w(x))=\lim_{x
ightarrow a}u(x)\pm\lim_{x
ightarrow a}v(x)\pm\lim_{x
ightarrow a}w(x)\) and the power rule \(\lim_{x
ightarrow a}x^{n}=a^{n}\).
\(\lim_{x
ightarrow1}(3x^{4}-3x^{2}+6)=3\lim_{x
ightarrow1}x^{4}-3\lim_{x
ightarrow1}x^{2}+\lim_{x
ightarrow1}6\)
\(=3\times1^{4}-3\times1^{2}+6\)
\(=3 - 3+6=6\)

Since \(6x\leq g(x)\leq3x^{4}-3x^{2}+6\) for all \(x\), and \(\lim_{x
ightarrow1}6x=\lim_{x
ightarrow1}(3x^{4}-3x^{2}+6) = 6\)

Answer:

\(6\)